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====== Time Delay ======

Some continuous-time models incorporate a time-delay in the rate equation, to account for processes that take a substantial amount of time to complete (such as gestation).  For example, a gestation period of $\tau$ in a population with density $n$ might be be represented as
\[
  \diff{}{t}n(t) = \beta n(t-\tau).
\]

There are various ways to capture the idea of a delay in a reaction-kinetics framework:

====== Intermediate state ======

We could just make the intermediate stage explicit.  For example, for gestation instead of just treating individuals as indistinguishable, $N$, we subtype them into $F$=fertile and $G$=gestating types.  In the simplest birth processes we could then have
\[
\begin{array}{rcl}
  F & \xrightarrow{\pi} & G \\
  G & \xrightarrow{\beta} & 2 F.
\end{array}
\]
In this example, the mass action dynamics would be
\[
\begin{array}{rcl}
  \diff{f}{t} & = & 2 \beta g - \pi f \\
  \diff{g}{t} & = & \pi f - \beta g \\
  \diff{n}{t} & = & \beta g
\end{array}
\]
where $n=f+g$.

In some (more subtle) cases a separation of timescales analysis allows a closed-form expression, $\diff{n}{t} = h(n)$, for some function $h$.

====== Taylor expansion ======

Another approach is mathematically inspired.  We begin with a lowest-order Taylor expansion of $n(t-\tau)$ around $t$:
\[
  n(t-\tau) \approx n(t) - \tau \diff{}{t} n(t).
\]
Then we simply plug-in the rate equation for $\diff{n}{t}$ and rearrange to find a closed-form expression.  For example,
\[
\begin{array}{rcl}
  \diff{}{t}n(t) & = & \beta n(t-\tau) \\
  \diff{}{t}n(t) & = & \beta \left[ n(t) - \tau \diff{}{t} n(t) \right] \\
  \diff{n}{t} \left[ 1 + \beta \tau \right] & = & \beta n \\
  \diff{n}{t} & = & \frac{\beta}{ 1 + \beta \tau } n.
\end{array}
\]
If we can write the rate equation as a polynomial of the densities then we can extract reactions that would give rise to it.  In this case,
\[
  N \xrightarrow{\beta/(1 + \beta \tau)} 2 N
\]
we find a simple birth process with a reduced birthrate.